Math 4710/5710 (Spring 2025) Final Project Information


Selected Topics:
 
Final Project & Presentation Guidelines:

Instead of a Final Exam, we will have a final "project" of sorts. Everyone should select a topic somehow related to this class (suggestions below). This should be something that we didn't cover or at least something we didn't cover in detail.

  • Pick and topic and clear it with me (in person or via email).
  • Study your topic.
  • Create a handout. This should be at least one page front and back.
    • This should be typed up nicely. While I prefer LaTeX, other word processing software (like Word) is ok too.
    • Your classmates are your audience. (Use class background.)
    • Let us know where to go for further reading.
      [Cite any valuable resources - like websites and textbooks.]
    • You might want to give a brief historical blurb.
    • Would worked out examples help us understand your topic?
         If so, include a few.
    • Is your topic about a "big theorem"?
         Include its proof or a sketch of its proof.
  • You will give a presentation during the final exam period: Wednesday, May 7th from 8am until 10:30am.
    • Plan on 5 to 10 minutes.
    • Your presentation could use slides (like tex or powerpoint)
      or you could just write on the board.
  • Email me s copy of your handout by Wednesday, May 7th at 8am.

Sample Slides:
• My Fall 2020 MAT 2110 page has some sample slides.
• Fuchs' Problem Slides [Source: (.tex)]
• Infinite Talk Slides [Source: (.zip)]

Example Handouts:
• Why not model after one of my monstrosities? [Posted below.]
• Examples where I cite some reasources:
   → Differential Algebra and Liouville's Theorem [Source: (.tex)]
   → Some differential Galois Theory [Source: (.tex)]

Random Topic Ideas:
  • Just look at Munkres' Table of Contents and pick a section that we skipped/didn't get to
  • More Set Theory - independence results or forcing or ordinals or
    Martin's Axiom or the Diamond Principle
  • Construct various counter-examples to demonstrate things like "not locally connected" or "sequentially compact but not compact" etc.
    Ex: The long line or broom spaces or ...
  • When do sequences determine the elements in the closure?
  • Suslin's problem
  • More about separation like building weird examples or results like Urysohn's Lemma or his metrization theorem or the Tietze Extension Theorem ...
  • Ascoli's Theorem
  • More about compactifications like the Stone-Cech Compactification
  • Rings of continuous functions
  • More on quotients/gluing
  • Various types of connectedness - how are they related?
  • Various types of compactness - how are they related?
  • Category Theory
  • More about Algebraic Topology:
    More about the fundamental group or Homology or Cohomology or ...
  • Knot Theory
  • Invariance of domain and dimension
  • Metrization theorems
  • What is a (topological) manifold? Or topological group?
  • Riemann surfaces
  • Jordan Curve Theorem
  • Banach-Tarski Theorem
  • Space filling curves